Exploring node interaction relationship in complex networks by using high-frequency signal injection

Many practical systems can be described by complex networks. These networks produce, day and night, rich data which can be used to extract information from the systems. Often, output data of some nodes in the networks can be successfully measured and collected while the structures of networks producing these data are unknown. Thus, revealing network structures by analyzing available data, referred to as network reconstruction, turns to be an important task in many realistic problems. Limitation of measurable data is a very common challenge in network reconstruction. Here we consider an extreme case, i.e., we can only measure and process the data of a pair of nodes in a large network, and the task is to explore the relationship between these two nodes while all other nodes in the network are hidden. A driving-response approach is proposed to do so. By loading a high-frequency signal to a node (defined as node $A$), we can measure data of the partner node (node $B$), and work out the connection structure, such as the distance from node $A$ to node $B$ and the effective intensity of interaction from $A$ to $B$, with the data of node B only. A systematical smoothing technique is suggested for treating noise problem. The approach has practical significance.

Figure 1

(a) Schematic figures of a complex network. The network is randomly constructed with link probability $p=0.10$. Red (solid) node 1 represents node $A$ to which high-frequency periodic driving could be applied, while blue (solid) node 16 indicates node $B$ of which output data could be measured and thus available. All other black (hollow) nodes are hidden and not observable. (b) Rearrangement of all nodes in network according to their distances from $A$. Unlike panel (a), here only part of the interactions, i.e., the shortest link paths from $A$ to the given nodes, are presented. Blue (colored) links show the interaction path from $A$ to $B$.

Figure 2

Results of $G_{\mu }-\omega$ computed from data produced by linear network (13). Four nodes 12, 6, 3, 2 [panels (a)–(d), respectively] each with different dominant distances $d=1,2,3,4$ from node $A$ are considered in the picture. Various symbols (cross, triangle, square, five-point star, and asterisk) show the numerical computation results from available data $x_B\left(t\right)$, and solid curves are given by the theoretical predictions from Eq. (11) with $I_{BA}^{\left(d\right)}$ shown in Table 1, columns L. It is convenient to find that $G_{\mu =d+1}$ curves computed by Eq. (9) are $\omega$ independent. Both lines coincide with each other perfectly. For all nodes demonstrated here we have $d={\nu }_s$, and ${\tilde{I}}_{BA}^{\left(d\right)}\approx I_{BA}^{(d={\nu }_s)}$, where ${\nu }_s$ represents the structurally shortest distance of interaction from node $A$ (comparing the bold quantities in Table 1, columns L and R).

Figure 3

Simulation results of Lorenz oscillators Eq. (14) for $\sigma =2, p=5.6, q=s=0.2, a=0.2, b=8/15, D=1, \omega =20\pi$. The fixed coupling matrix ${\widehat{\mathbf{\Phi }}}^{\prime }$ is given in Fig. 13 in appendix. Network dynamics are strongly chaotic in both time and space. (a) Time sequences of node $A=1$. (b) Time sequences of node $B$ (node 4). The two curves present data produced with and without periodic driving in node $A$. Trajectories in variable planes of (c) $z_1-x_1$ and (d) $z_4-x_4$ with periodic driving applied. The same as panel (d) but with data of (e) the first-order and (f) the third-order derivatives of variables $x$ and $z$ over $t$. (g)–(j) The same as panels (c)–(f) but without driving at node $A$. Network systems (14) with or without driving are both chaotic, and the curves in panels (a) and (b) deviate quickly with time. Nevertheless, the chaotic attractors are stable against driving signal perturbation, so that the attractors in panels (c)–(f) (with driving) are similar to those of panels (g)–(j) (without driving), respectively.

Figure 4

(a)–(d) Calculation results of $G_{\mu }-\omega$ with all data collected by running Eq. (14) with $D=50$. Results of four nodes 5, 7, 4, 23 [panels (a)–(d), respectively] with distances $d=1,2,3,4$ are shown in the picture. The agreement of data-based numerical results with the theoretical prediction of Eq. (11) is fairly good, although the network dynamics are strongly chaotic. Again for all nodes demonstrated, $d={\nu }_s$, and $I_{BA}^{(d={\nu }_s)}\approx {\tilde{I}}_{BA}^{\left(d\right)}$. (e), (f)The same as panels (a) and (d) without driving [$D=0$ in Eq. (14)]. The curves in panels (e) and (f) are much lower than those in panels (a) and (d) in the high-frequency domain, so the resonant behavior of the autonomous network appearing in the small-frequency region do not effectively influence the reconstruction results.

Figure 5

Calculation results of $G_{\mu }-\omega$ for GRN, Eqs. (15) and (16). Four nodes 11, 4, 26, 7 with distances $d=1,2,3,4$ [panels (a)–(d), respectively] are considered in the picture. (a)–(d) The shortest distances of the given nodes $d={\nu }_s$ and the corresponding experimental average intensities are depicted correctly. (e), (f) Similar to panels (a) and (d) without driving at node $A$. The response values are much lower than those in panels (a) and (d) and do not affect reconstruction results.

Figure 6

This figure shows some rare cases in Table 3. All lines contain the same information as in Fig. 5. In panels (a)–(c) solid curves are given again by Eq. (11), however, with dominant interaction intensities $I_{BA}^{(d={\nu }_s+1)}$ [in panels (a) and (b)], and $I_{BA}^{(d={\nu }_s+2)}$ [in panel (c)], but not $I_{BA}^{(d={\nu }_s)}$ as in Fig. 5. Now it is not the interactions of the shortest distances ${\nu }_s$ but some interactions of farther distances $d>{\nu }_s$ dominating the links, because the shortest interactions are too weak. In panel (d) there is no $\omega$-independent $G_{\mu }$ curve, and thus no single interaction is dominant. We need to reconstruct $|{\tilde{I}}_{BA}^{\left(d_i\right)}|, i=1,2,...,r$ from Eq. (12). By using the method stated in context we reconstruct two dominant interaction distances $d_1=3, d_2=4$, shown in Table 3, columns R (bold numbers for node 32), which coincide approximately with the actual structural ones in Table 3, columns L for the same node.

Figure 7

(a) Network with chain structure. Signal ($D=10$) is applied to the first colored node, and data of two nodes $B_1=7$ and $B_2=11$ with $d_1=6, d_2=10$ are measured for reconstruction. For reconstruction of long distances, high-precision measurement data (about 20 significant digits) are necessary. (b)–(e) results of $G_{\mu }-\omega$ for chain network. (b), (c) Same form of equations as Eq. (13) is applied for the simulation, but with different network links. The chain is connected by unidirectional links ${\mathrm{\Phi }}_{i+1,i}=1$; otherwise ${\mathrm{\Phi }}_{i,j\ne i}=0$. Strong dissipations of ${\mathrm{\Phi }}_{ii}=-3, i=1,2,...,N$ are applied. (d), (e) Similar equations as Eq. (14) are applied for the simulation. For studying the validity of the method for different interaction structures, here we use bidirectional links ${\mathrm{\Phi }}_{ij}=1$ for $i=j+1$; ${\mathrm{\Phi }}_{ij}=0.9$ for $i=j-1$, and ${\mathrm{\Phi }}_{ij}=0$ for others. In all cases above the method of interaction reconstruction works very well for considerably long distances.

Figure 8

The same images as in Figs. 3–3, but with different data produced by Eq. (17), where noise of intensity $Q={10}^{-4}$ is applied. Although small noise does not introduce considerable changes in (a), (b) time sequences and (c), (d) variable trajectories and (e) its first-order derivative, noise-induced fluctuations can be greatly enhanced as (f) the order of derivative increases, which makes the reconstruction task for nodes with long distances difficult.

Figure 9

The same as Fig. 4 but with data generated by Eq. (17). In addition, we apply a longer emulation time $T=10000$ and a signal amplitude $D=1$. High-order derivatives greatly enhance the noise effects, which can be much larger than the theoretical predicted signal amplitudes and make reconstruction analyses confused. Particularly, no trace of node relationships can be seen for nodes with large distances from node $A$; for example, in panels (c) and (d) we cannot seek out $d=3$ and 4.

Figure 10

The same as Fig. 9 but with the systematical smoothing algorithm (18) and (19) applied. The results of Figs. 4–4 are well restored in spite of the presence of noise. The smoothing computations greatly reduce the noise disturbances and we can correctly reconstruct the dominant distances and the corresponding interaction intensities of the given nodes with satisfactory approximation.

Figure 11

Error curves of Eq. (20) calculated from data of node 7 in Eq. (14), plotted (a) vs $D$ ($\omega =20\pi ,T=1000$), (b) vs $\omega$ ($D=10,T=1000$), and (c) vs $T$ ($D=10,\omega =20\pi$), respectively. Error results of node 26 in Eqs. (15) and (16) plotted (d) vs $D$ ($\omega =20\pi ,T=1000$), (e) vs $\omega$ ($D=10,T=1000$), and (f) vs $T$ ($D=10,\omega =40\pi$), respectively. (g)–(i) The same as above but with data calculated from node 23 in Eq. (17a), considering noise effects. The abnormal valley-like behavior in panel (d) and (h) is explained in the related context.

Figure 12

Structural matrix $\left({\widehat{\mathrm{\Phi }}}^{\prime }\right)$ in network of Eq. (13), which is shown in color coded graphical form. The simple network is randomly constructed. Probability for any link to be nonzero reads $p=0.1$. If $j$ interacts with $i, {\mathrm{\Phi }}_{ij}^{\prime }$ has equal possibility to be 1 or $-1$. To maintain the network dynamics stable, we use sufficiently large dissipation as ${\mathrm{\Phi }}_{ii}^{\prime }=-3,i=1,2,...,N$. Apart from dissipation terms, there are 83 positive links (1) and 78 negative links ($-1$) in this matrix.

Figure 13

Structural matrix $\left({\widehat{\mathrm{\Phi }}}^{\prime }\right)$ used in networks (14) and (17a), which is shown in color-coded graphical form. This network is also randomly constructed with link probability $p=0.1$. If there is no link from $j$ to $i, {\mathrm{\Phi }}_{ij}^{\prime }=0$. If $j$ interacts $i, {\mathrm{\Phi }}_{ij}^{\prime }$ has equal possibility to be positive or negative, and we randomly choose its absolute value in a range from 0.8 to 1.2.

Figure 14

Structural matrix $\left(\widehat{f}\right)$ used in Eq. (15), which is shown in color coded graphical form. Network with a system size $N=40$ is constructed and for each node five inputs from other nodes are randomly chosen. There are five nonzero numbers in each row. The nonzero element $f_{ij}$ in this matrix represents there is connection from nodes $i$ to $j$. Each input link has half chance to be active ($f=1$) or repressive ($f=-1$).